Research Article | Open Access
A New Method for Research on the Center-Focus Problem of Differential Systems
We will introduce Mironenko’s method to discuss the Poincaré center-focus problem, and compare the methods of Lyapunov and Mironenko. We apply the Mironenko method to discuss the qualitative behavior of solutions of some planar polynomial differential systems and derive the sufficient conditions for a critical point to be a center.
As we know [1, 2], for polynomial differential system where and are real constants, there has been a longstanding problem, called the Poincaré center-focus problem; for the system (1) find explicit conditions of and under which (1) has a center at the origin ; that is, all the orbits nearby are closed. The problem is equivalent to an analogue for a corresponding periodic equation to have periodic solutions. To see this let us note that the phase curves of (1) near the origin in polar coordinates , are determined by (2), where and are polynomials in and .
Since the closed orbits of (1) correspond to -periodic solutions of (2), the planar vector field (1) has a center at (0, 0) if and only if (2) has a center at ; that is, all the solutions nearby are periodic: [1, 2].
To discuss the center-focus problem, there are Lyapunov’s method and the others; see the works of Z. Zhang and so forth [2–8]. One of the most commonly used methods is Lyapunov’s method. However, sometimes, the use of this method to calculate the focus quantities is very difficult and the amount of calculation is very large.
In this paper, we apply the method of Mironenko  (reflecting function method) to study the qualitative behavior of solutions of (2) and give the sufficient conditions for to be a center. We give a few examples to illustrate that, sometimes, Mironenko's method is better than Lyapunov's method.
In the present section, we introduce the concept of the reflecting function, which will be used throughout the rest of this paper.
Consider differential system which has a continuously differentiable right-hand side and with a general solution . For each such system, the reflecting function is defined (see Mironenko's ) as . Therefore, for any solution of (3), we have , .
If system (3) is -periodic with respect to , and is its reflecting function, then is the Poincaré mapping of (3) over the period . Thus, the solution of (3) defined on is -periodic if and only if is a fixed point of . The stability of this periodic solution is equivalent to the stability of the fixed point .
A differentiable function is a reflecting function of system (3) if and only if it is a solution of the Cauchy problem
If is a reflecting function of system (3), then it is also reflecting function of the following systems: where is an arbitrary vector function. Thus, the shift operators [9, 10] on interval for such systems coincide, and the qualitative behaviors of the periodic solutions of these systems are the same. See papers [11–18] in which reflecting functions are also applied to study the qualitative behavior of solutions of some nonlinear and nonautonomous differential systems.
In the following, we always assume that all equations in this paper have a continuously differentiable right-hand side and have a unique solution for their initial value problem.
2. Main Results
Now, let us consider differential equation (2).
Lemma 2 ([see ). If all the solutions of (2) are -periodic, then its reflecting function is -periodic, too.
First, we introduce the Mironenko’s method. Suppose that is the reflecting function of (2), where . Substituting it into and equating the coefficients of like powers of , we get the following infinite equations If , then and is a center. If this is a and , then , where is a constant and when , is an unstable focus; when , is a stable focus.
By the above description, it seems that Mironenko's method is similar to Lyapunov's method, but do not forget that reflecting function has some good symmetry properties, which make our calculations greatly reduced. The following example will illustrate this advantage.
Example 3. Consider system Taking , it is transformed to equation
Equating the coefficients of like powers of implies Solving it we get It implies so
From above, we see that the calculation of the expression of will become more difficult. Thus, it is not easy to judge is a center?
Now, we try to use the method of Mironenko to discuss is a center? In (7) taking we get Equating the coefficients of like powers of implies and so forth. Now we can conjecture that ; that is, . It is not difficult to check that is a solution of (7), so it is the reflecting function of (10). As , so the origin point of system (9) is a center.
Such example shows us that, sometimes, the method of Mironenko is better than Lyapunov’s method. Just this advantage is not enough. By the equivalence, we know that also is the reflecting function of the following equations: where is an arbitrary function. If we take where is a positive integer, then (20) becomes Taking , then (22) is transformed to and its origin point is a center too.
Thus, by using the method of Mironenko we not only solve a center-focus problem, but at the same time open a class of differential equations with the same character of point . Therefore, we can say, sometimes, the method of Mironenko is more effective than Lyapunov’s method.
Theorem 5. If there is a function , which satisfies where . Then all the solutions of (2) determined on are -periodic; that is, is a center.
Proof. Let be the reflecting function of (2). Now, we prove that it satisfies .
In fact, by relation (24), for any solution of (2), we have In view of , we have [9, page 65] that is, which implies that is a -periodic function with respect to . By Lemma 1, the proof of the present theorem is finished.
Corollary 6. If there is a differentiable -periodic function such that
are -periodic and odd functions, then all the solutions defined on of equation
This result is implied by Theorem 5 if we take and put
Example 7. Taking , we get that all the solutions determined on of the Riccati equation are -periodic.
Theorem 8. For (2), suppose that , , . Then is a center.
Proof. By the assumptions, it is not difficult to check that is the reflecting function of (2), so is a center.
Theorem 9. For (2), suppose that and where , , . Then all the solutions of (2) and the following equations determined on are -periodic; that is, is a center, where is a continuously differentiable -periodic function with respect to and .
Theorem 10. Suppose that for (2) is a center and functions are -periodic with respect to and satisfy Then for the following equations is a center, too. Here are arbitrary continuously differentiable -periodic odd functions.
Proof. By the assumptions and [9, 14], we know that the reflecting functions of (2) and (36) coincide and all the solutions of (2) determined on are -periodic. By Lemma 2, we know that the common reflecting function is -periodic; thus all the solutions of (36) determined on are -periodic; that is, is a center, too.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The work supported by the NSF of Jiangsu of China under Grant no. BK2012682 and the NSF of China under Grant nos. 11271026, 61374010.
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